#8634819

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(2,5)(6,7)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(11,18)(14,17)(19,24)(22,23)(26,29)(32,34)(33,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,29,31,26)(27,32,30,34)(28,36,38,33)(35,39,37,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,27)(26,32,29,34)(28,35,38,37)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8542015);